Heat kernel coefficients on Kahler manifolds

TL;DR

This paper derives explicit combinatorial and graph-theoretic formulas for heat kernel coefficients on Kähler manifolds, extending Polterovich's work on Riemannian manifolds to the complex setting.

Contribution

It introduces a novel combinatorial approach and explicit formulas for heat kernel coefficients on Kähler manifolds, expanding the theoretical understanding of heat kernel asymptotics.

Findings

Derived a combinatorial formula for powers of the complex Laplacian.

Established an explicit graph-theoretic formula for heat coefficient numerics.

Extended Polterovich's Riemannian results to Kähler manifolds.

Abstract

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of K\"ahler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich's formula.